Question

The energy levels of a quantum harmonic oscillator are:

• A. Degenerate
• B. Equally spaced
• C. Proportional to \(n^2\)
• D. Continuous

Hint:

For standard quantum systems, keep their level spacings in mind: - Quantum Harmonic Oscillator: Energy levels are equally spaced (\(\Delta E = \hbar\omega\)). - Infinite Square Well (Particle in a box): Spacing increases with higher values of \(n\) (\(E_n \propto n^2\)). - Hydrogen Atom (Coulomb Potential): Spacing decreases as \(n\) goes up (\(E_n \propto -\frac{1}{n^2}\)).

Answer 1

The Correct Option is B

Concept: The quantum harmonic oscillator is a fundamental system in quantum mechanics used to model vibrations. The potential energy function for a particle of mass \(m\) vibrating with an angular frequency \(\omega\) is given by the parabolic function \(V(x) = \frac{1}{2}m\omega^2 x^2\). Solving the time-independent Schrödinger equation for this potential yields quantized, discrete energy eigenvalues.

Step 1: Analyzing the energy eigenvalue expression.
The allowed energy levels for a one-dimensional quantum harmonic oscillator are governed by the following formula: \[ E_n = \left(n + \frac{1}{2}\right)\hbar\omega \] where:
• \(n\) is the principal quantum number, which must be a non-negative integer: \(n = 0, 1, 2, 3, \ldots\)
• \(\hbar\) is the reduced Planck constant (\(\hbar = \frac{h}{2\pi}\)).
• \(\omega\) is the characteristic angular frequency of the oscillator.

Step 2: Determining the spacing between adjacent energy levels.
To establish how the levels are distributed, let us determine the energy difference (\(\Delta E\)) between any two consecutive energy states, \(E_n\) and \(E_{n+1}\): \[ \Delta E = E_{n+1} - E_n \] Substituting the formula for both levels: \[ E_{n+1} = \left((n + 1) + \frac{1}{2}\right)\hbar\omega = \left(n + \frac{3}{2}\right)\hbar\omega \] \[ E_n = \left(n + \frac{1}{2}\right)\hbar\omega \] Now, computing the difference: \[ \Delta E = \left(n + \frac{3}{2}\right)\hbar\omega - \left(n + \frac{1}{2}\right)\hbar\omega \] Factoring out the common constant term \(\hbar\omega\): \[ \Delta E = \left[\left(n + \frac{3}{2}\right) - \left(n + \frac{1}{2}\right)\right]\hbar\omega \] \[ \Delta E = \left(n - n + \frac{3}{2} - \frac{1}{2}\right)\hbar\omega = 1 \cdot \hbar\omega = \hbar\omega \] Since \(\Delta E = \hbar\omega\) is completely independent of the quantum number \(n\), the separation between any two successive energy levels remains perfectly constant. Therefore, the energy levels are completely discrete and uniformly or equally spaced by a fixed increment of \(\hbar\omega\).

Step 3: Verification of other options.

• Option (1) Degenerate: In a one-dimensional system, each unique energy value corresponds to exactly one unique, non-degenerate quantum state. Thus, they are non-degenerate.
• Option (3) Proportional to \(n^2\): The energy varies linearly with \(n\), not as a quadratic function of \(n\) (unlike a particle trapped in an infinite potential square well where \(E_n \propto n^2\)).
• Option (4) Continuous: Because \(n\) is strictly restricted to integer values, the permitted energy states are bounded into a discrete ladder rather than being continuous. Consequently, option (2) is mathematically accurate.